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Just to add to that, so does both Columbia's Financial Engineering and Operations Research programs. I know (not directly though) that one guy, not in my year but an earlier year, graduated from the financial engineering program and did his PhD in Princeton's ORFE program. In my year, two women, both graduates of the operations research program, are doing their PhDs: one in MCF at Oxford and another in ORFE at Princeton.Columbia's Financial Econ, Stanfords ICME, and Princeton's MFIN all usually have a couple students pursuing PhDs post graduation (if not directly transitioning to one at the same school), just don't know anything about MIT
From what I've seen, in general, I think as long as you came from a reputable MFE program (from a reputable school--generally top US school) that allows you flexibility to take more research-oriented courses, have research experience (publications are obviously the best, as they show that you can actually produce quality research), and can show you can tackle PhD level core courses (on par with those from whichever top PhD programs you'll interested in ofc) or have a strong quantitative background, then you're fine. I know Columbia Business School PhDs must take some of the same core PhD courses as Columbia IEOR PhDs, but not sure about the exact requirements.
I am curious whether MIT MFin students apply for MIT ORC's PhD program (some faculty there have joint appointments with Sloan, e.g., prof. Andrew Lo). I really like that program and their students land excellent industry/academic jobs. I know quite a few professors in Columbia IEOR who graduated from there.